Proposition: Deterministic expectation-label obstruction

A channel with fixed auxiliary input cannot output a deterministic classical label $\ell (\varrho )$ for every state on the full convex state space if at least two labels occur. The impossibility remains if the matter output is additionally required to equal the input.

A circuit acting only on classical code registers, and as identity on internal matter, cannot make a fixed code input's output depend on an unencoded matter expectation.

Proof. Choose states assigned distinct labels ${\ell }_1,{\ell }_2$. Linearity forces the label marginal on their equal mixture to be

which is not a deterministic label. The additional identity requirement on matter cannot remove this contradiction.

For a fixed code input, a code-only circuit has the same code output for every separate matter state. Therefore it cannot distinguish expectations not represented in that input.